Free Statistics Help Book
An Interactive Multimedia introductory-level statistics book.
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Chapter
Section
Introduction :  

Summation Notation



Prerequisite
None


Learning Objectives
1. Use summation notation to express the sum of all numbers
2. Use summation notation to express the sum of a subset of numbers
3. Use summation notation to express the sum of squares


Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation.


Let’s say we have a variable X that represents the weights (in grams) of 4 grapes. The data are shown in Table 1.

Table 1. Weights of 4 grapes.

Grape X
1

2

3

4
4.6

5.1

4.9

4.4



We label Grape 1′s weight X1, Grape 2′s weight X2, etc. The following formula means to sum up the weights of the four grapes:





The Greek letter capital sigma (Σ) indicates summation. The “i = 1″ at the bottom indicates that the summation is to start with X1 and the 4 at the top indicates that the summation will end with X4. The Xi indicates that X is the variable to be summed as i goes from 1 to 4. Therefore,


= X1 + X2 + X3 + X4 = 4.6 + 5.1 + 4.9 + 4.4 = 19.0.


The symbol





indicates that only the first 3 scores are to be summed. The index variable i goes from 1 to 3.


When all the scores of a variable (such as X) are to be summed, it is often convenient to use the following abbreviated notation:





Thus when no values of i are shown, it means to sum all the values of X.


Many formulas involve squaring numbers before they are summed. This is indicated as


ΣX²= 4.62 + 5.12 + 4.92 + 4.42 = 21.16 + 26.01+ 24.01 + 19.36 = 90.54


Notice that:





because the expression on the left means to sum up all the values of X and then square the sum (19² = 361) whereas the expression on the right means to square the numbers and then sum the squares (90.54, as shown).


Some formulas involve the sum of cross products. Table 2 shows the data for variables X and Y. The cross products (XY) are shown in the third column. The sum of the cross products is 3 + 4 + 21 = 28.

Table 2. Sums of Cross Products.

X Y XY
1

2

3
3

2

7
3

4

21



In summation notation, this is written as: ΣXY = 28.

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